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## Dirichlet Distribution

Let be parameters. Then we can define the pdf

over the simplex

This is called the Dirichlet distribution.

Multinomial distribution. Suppose that represents the probability for the -th outcome for each , and outcomes are repeatedly obtained according to the probability . Then we can count for each the number of the -th outcome, and the chance of getting the values is given by the multinomial distribution

over

Conjugate family of Dirichlet distribution. Let be the prior density for with hyperparameter . Given the data from the multinomial distrubution with parameter , we can obtain the posterior density

which is the Dirichlet distribution .